We study a compound Poisson random field on plane and examine its various fractional variants. We derive the distributions of these random fields and in some particular cases, obtain their associated system of governing differential equations. Additionally, various weak convergence results are derived. Also, we introduce and study some fractional variants of the Poisson random field (PRF) that includes the space fractional and the space-time fractional Poisson random fields. We discuss various distributional properties for the fractional PRFs and obtain their time-changed representations. Lastly, we define and study some fractional compound Poisson random fields using the fractional variants of PRF.